where \(b = 3 - \frac{(1+r+s)^{2}}{2k+1}\) , \(c = -1- \frac{(1+r-s)(1+r+s)}{2k+1}\)  and \(d = 3 - \frac{(-1-r+s)^{2}}{2k+1}\) .
Applying Lemma REF  to the sub-matrix complementary to the last row and column with \(a = 2k-2\)  and \(b, c,d \)  as given,
with simplification performed in MAGMA [1]}, we obtain the following factorisation of the determinant:
\( ( 48k^3 - 8k^2 r^2 - 16 k^2 r - 12k^2 s^2 - 24 k^2 + 4k r^2 + 8kr + 4ks^2 - 8k + 4 ) (2k-2)^{2k-2}.\) 
